321 research outputs found
On the 2-extendability of planar graphs
AbstractSome sufficient conditions for the 2-extendability of k-connected k-regular (k⩾3) planar graphs are given. In particular, it is proved that for k⩾3, a k-connected k-regular planar graph with each cyclic cutset of sufficiently large size is 2-extendable
Hamiltonian chordal graphs are not cycle extendible
In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle
extendible; that is, the vertices of any non-Hamiltonian cycle are contained in
a cycle of length one greater. We disprove this conjecture by constructing
counterexamples on vertices for any . Furthermore, we show that
there exist counterexamples where the ratio of the length of a non-extendible
cycle to the total number of vertices can be made arbitrarily small. We then
consider cycle extendibility in Hamiltonian chordal graphs where certain
induced subgraphs are forbidden, notably and the bull.Comment: Some results from Section 3 were incorrect and have been removed. To
appear in SIAM Journal on Discrete Mathematic
On Cyclic Edge-Connectivity of Fullerenes
A graph is said to be cyclic -edge-connected, if at least edges must
be removed to disconnect it into two components, each containing a cycle. Such
a set of edges is called a cyclic--edge cutset and it is called a
trivial cyclic--edge cutset if at least one of the resulting two components
induces a single -cycle.
It is known that fullerenes, that is, 3-connected cubic planar graphs all of
whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this
article it is shown that a fullerene containing a nontrivial cyclic-5-edge
cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces
whose neighboring faces are also pentagonal. Moreover, it is shown that has
a Hamilton cycle, and as a consequence at least perfect matchings, where is the order of .Comment: 11 pages, 9 figure
The polynomial method for 3-path extendability of list colourings of planar graphs
We restate Thomassen's theorem of 3-extendability, an extension of the famous
planar 5-choosability theorem, in terms of graph polynomials. This yields an
Alon--Tarsi equivalent of 3-extendability
Drawing Planar Graphs with a Prescribed Inner Face
Given a plane graph (i.e., a planar graph with a fixed planar embedding)
and a simple cycle in whose vertices are mapped to a convex polygon, we
consider the question whether this drawing can be extended to a planar
straight-line drawing of . We characterize when this is possible in terms of
simple necessary conditions, which we prove to be sufficient. This also leads
to a linear-time testing algorithm. If a drawing extension exists, it can be
computed in the same running time
On the structure of the directions not determined by a large affine point set
Given a point set in an -dimensional affine space of size
, we obtain information on the structure of the set of
directions that are not determined by , and we describe an application in
the theory of partial ovoids of certain partial geometries
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